A394857 Lexicographically earliest sequence of distinct positive integers such that the prime tower factorizations of two consecutive terms have no common prime number.

1, 2, 3, 4, 5, 6, 7, 8, 11, 9, 13, 10, 17, 12, 19, 14, 15, 16, 21, 20, 23, 18, 29, 22, 27, 25, 31, 24, 35, 26, 33, 28, 37, 30, 41, 32, 39, 34, 43, 36, 47, 38, 51, 44, 53, 40, 59, 42, 55, 46, 57, 49, 61, 45, 67, 48, 65, 54, 71, 50, 69, 52, 73, 56, 79, 58, 77

Offset

1,2

Comments

The prime tower factorization of a number is defined in A182318.

This sequence is a permutation of the positive integers with inverse A394859:

- we can always extend the sequence with some prime number larger than all prior terms,

- for any prime number p, the first term >= p is necessarily p,

- all prime numbers appear in the sequence, in natural order,

- each time we hit a prime number p, the least missing composite number c < p if any must follow immediately, so eventually every composite number will appear in the sequence.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, PARI program

Index entries for sequences that are permutations of the natural numbers

Formula

gcd(A336965(a(n)), A336965(a(n+1))) = 1.

Example

The first terms are:
  n   a(n)  a(n)-th row of A336964
  --  ----  ----------------------
   1     1  [1]
   2     2  [2]
   3     3  [3]
   4     4  [2]
   5     5  [5]
   6     6  [2, 3]
   7     7  [7]
   8     8  [2, 3]
   9    11  [11]
  10     9  [2, 3]
  11    13  [13]
  12    10  [2, 5]
  13    17  [17]
  14    12  [2, 3]
  15    19  [19]
  16    14  [2, 7]
  17    15  [3, 5]
  18    16  [2]
  19    21  [3, 7]
  20    20  [2, 5]

Prog

(PARI) \\ See Links section.

Crossrefs

Cf. A182318, A336964, A336965, A394855, A394858, A394859 (inverse).

Sequence in context: A354435 A358277 A361513 * A255004 A257728 A377193

Adjacent sequences: A394854 A394855 A394856 * A394858 A394859 A394860

Keyword

nonn

Author

Rémy Sigrist, Apr 04 2026

Status

approved